Seems to me that Rookie League ballparks would be ideal in many ways. FL or AZ. They don't start up until late June.
I don't think those would work. They are not set up for fans to attend. Also, they actually are being used at the time of the D3 CWS with a number of players who are on the DL and also those who are in "extended" ST (did not make a roster for a full season). Separate from the facility issue would be having a group to do all the work to put on the CWS.
Finally, those fields don't have press box facilities which would meet NCAA requirements.
What might work would be places which have short season teams, like the NY/PENN league or NWL.
While a bit of a challenge for travel from the South and West, a place like Doubleday field in Auburn NY makes sense. Can fly into Syracuse, close to the HOF, awfully nice facility seating about 2,500 and very, very good community support, at least for the Doubleday short season team. Aberdeen MD might be another similar type of option from the NY/PENN league.
When travel is considered, though, I would tend to think the NCAA will look toward the Mid-West, where they won't find short season fields available. Some really great facilities in the MWL (South Bend, Peoria, Lansing, etc) but they may run into similar use issues as Appleton.